Our main result is a local limit law for the empirical spectral distribution of the anticommutator of independent Wigner matrices, modeled on the local semicircle law. Our approach is to adapt some techniques from recent papers of Erdös-Yau-Yin. We also use an algebraic description of the law of the anticommutator of free semicircular variables due to Nica-Speicher, the linearization trick due to Haagerup-Schultz-Thorbjørnsen in a self-adjointness-preserving variant and the Schwinger-Dyson equation. A by-product of our work is a relatively simple deterministic version of the local semicircle law.
|Original language||English (US)|
|Number of pages||33|
|Journal||Annales de l'institut Henri Poincare (B) Probability and Statistics|
|State||Published - Aug 1 2015|
Bibliographical notePublisher Copyright:
© 2015 Association des Publications de l'Institut Henri Poincaré.
- Linearization trick
- Local semicircle law
- Schwinger-Dyson equation
- Wigner matrices